A triangulated category is an additive category with an invertible shift functor and a specified class of distinguished triangles satisfying the triangle axioms. These require closure under isomorphism and rotation, completion of morphisms to triangles, compatibility of maps of triangles, and the octahedral axiom for compositions. Derived categories and the stable homotopy category are examples. In the latter, distinguished triangles arise from cofiber sequences of spectra. A thick subcategory is closed under shifts, these triangles, and retracts. Adding tensor-ideal closure is a further condition.
Triangulated Category
See also
Bridgeland Stability Condition, Cofiber, Derived Category, Kuznetsov Component, Semiorthogonal Decomposition, Serre Functor, Spherical Twist, t-Structure, Thick Subcategory, Thick Tensor Ideal, Tilted Heart, Topological SpectrumExplore with Wolfram|Alpha
References
The Stacks Project Authors. "The Definition of a Triangulated Category." The Stacks Project, Tag 05QK. https://stacks.math.columbia.edu/tag/05QK.Cite this as:
Weisstein, Eric W. "Triangulated Category." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TriangulatedCategory.html